549 



wliich shall be bisected by a given diameter ab, and shall also 

 be such that while it intersects a given chord CD in some point 

 E, its polar intersects the prolongation of that given chord, in 

 some other point f. 



9. The two sought points e, f, as being situated upon two 

 polars, are of course conjugate relatively to the surface ; they 

 are therefore also conjugate relatively to the chord cd, or, in 

 other words, they cut that given chord harmonically. The 

 four diametral planes abc, abe, abd, abf, compose therefore 

 an harmonic pencil ; the second being, in this pencil, har- 

 monically conjugate to the fourth ; and being at the same 

 time, on account of the polars, conjugate to it also with re- 

 spect to the surface, as one diametral plane to another. When 

 the ellipsoid becomes a sphere, the conjugate planes abe, abf 

 become rectangular ; and consequently the sought plane abe 

 bisects the angle between the two given planes abc and abd. 

 This solves at once the problem for the sphere ; for if, con- 

 versely, we thus bisect the given dihedral angle cabd by a 

 plane abe, cutting the chord cd in e, and if we take the har- 

 monic conjugate f on the same given chord prolonged, and 

 draw from e and f lines meeting ordinately the given diame- 

 ter ab, these two right lines will be situated in two rectangu- 

 lar or conjugate diametral planes, and will satisfy all the other 

 conditions requisite for their being polars of each other; but 

 each intersects the given chord cd, or that chord prolonged, 

 and therefore each intersects also, by (8), the polar of that 

 chord ; each therefore satisfies all the transformed conditions of 

 the problem, and gives a chord of solution, real or imaginary. 

 More fully, the ordinate ee' to the diameter ab, drawn from 

 the internal point of harmonic section e of the chord cd, 

 gives, when prolonged both ways to meet the surface, the 

 chord of real solution, Ps ; and the other ordinate ff' to the 

 same diameter ab, which is drawn from the external point of 

 section f of the same chord cd, and which is itself wholly ex- 

 ternal to the surface, is the chord of imaginary solution. But 



